# Effects of Vaccination Efficacy on Wealth Distribution in Kinetic Epidemic Models

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## Abstract

**:**

## 1. Introduction

## 2. Wealth Dynamics in Epidemic Phenomena

#### 2.1. The Kinetic Model

**Remark**

**1.**

#### 2.2. Evolution of Macroscopic Quantities

**Remark**

**2.**

## 3. Properties of the Kinetic Model

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

#### 3.1. Fokker–Planck Scaling and Steady States

**Remark**

**4.**

## 4. Numerical Results

#### 4.1. Test 1: Long-Time Behavior and Convergence to Equilibrium

- $\left(i\right)$
- ${\lambda}_{S}=0.15$, ${\lambda}_{I}=0.10$, ${\lambda}_{V}=0.30$, ${\lambda}_{R}=0.20$
- $\left(ii\right)$
- ${\lambda}_{S}=0.10$, ${\lambda}_{I}=0.05$, ${\lambda}_{V}=0.30$, ${\lambda}_{R}=0.15$

#### 4.2. Test 2: Wealth Inequalities and Vaccination Campaign

#### 4.3. Nonlinear Incidence Rate and Time-Varying Vaccine Efficacy

#### 4.3.1. Test 3A: ${\gamma}_{R}=0$

#### 4.3.2. Test 3B: ${\gamma}_{R}>0$

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Test 1. Comparison of the wealth distributions at the end of the epidemic for the kinetic system (2) with the explicit Fokker–Planck asymptotics (26) with scaling parameters $\u03f5=1,\frac{1}{2},{10}^{-3}$. (

**Left**) ${\lambda}_{S}=0.15$, ${\lambda}_{I}=0.10$, ${\lambda}_{V}=0.30$, ${\lambda}_{R}=0.20$. (

**Right**) ${\lambda}_{S}=0.10$, ${\lambda}_{I}=0.05$, ${\lambda}_{V}=0.30$${\lambda}_{R}=0.15$. In both cases we fixed $\overline{\beta}=0.2$, ${\gamma}_{I}=1/12$, $\alpha =0.005$, $\zeta =0.9$ and ${\sigma}^{2}=0.02$.

**Figure 2.**Test 2. Evolution of the epidemic dynamics from (9) for the choice of parameters $\overline{\beta}=0.15$, ${\gamma}_{I}=1/12$, $\alpha =0.01$ and $\zeta =0.95$ (

**left**), $\zeta =0.55$ (

**right**).

**Figure 3.**Test 2. Evolution of Gini index under the epidemic dynamics described in Figure 2 and for the choice of parameters ${\lambda}_{S}=0.10$, ${\lambda}_{I}=0.07$, ${\lambda}_{V}=0.30$, ${\lambda}_{R}=0.15$. Two vaccine efficacies were considered: $95\%$ (green) and $55\%$ (red). In both cases we considered ${\sigma}^{2}=0.02$.

**Figure 4.**Test 2. (

**Left**) evolution of the market risk ${\sigma}^{2}\left(t\right)$ as defined in (31) with $\mu =50$ and ${\sigma}_{0}^{2}=0.02$ in case of two different vaccine efficacies. (

**Right**) evolution of Gini index under the epidemic dynamics described in Figure 2 and epidemic-dependent market risk parameter (31).

**Figure 5.**Test 2. Time evolution of the wealth distribution of the kinetic model (2) in the scaling $\u03f5=5\times {10}^{-2}$ with vaccine efficacy $\zeta =0.55$ (

**left**column) or $\zeta =0.95$ (

**right**column) and with constant market risk ${\sigma}^{2}=0.02$ (top row) or ${\sigma}^{2}\left(t\right)$, defined in (31) with $\mu =50$. In all the evolutions we considered ${\lambda}_{S}=0.10$, ${\lambda}_{I}=0.07$, ${\lambda}_{V}=0.30$ and ${\lambda}_{R}=0.15$. The initial distribution was defined in (29) and (30). In the left image, we can observe the evolution of the wealth distribution for the kinetic model (2) in the scaling parameter $\u03f5=5\times {10}^{-2}$ with $\zeta =0.95$, whereas, in the right image we have the comparison between the behaviors of the Gini index with vaccine effectiveness, equal to $95\%$ (green line) and $65\%$ (red line). In both images we considered a variable market risk (31) with ${\sigma}_{0}^{2}=0.02$ and $\mu =50$ and ${\lambda}_{S}=0.10$, ${\lambda}_{I}=0.07$, ${\lambda}_{V}=0.30$ and ${\lambda}_{R}=0.15$.

**Figure 6.**Test 3. Wealth-dependent contact rate $\beta (w,{w}_{*})$ of the form (32) with $\overline{\beta}=8$, $c=7$, $\nu =2$.

**Figure 7.**Test 3A. Top row: epidemic dynamics with wealth-dependent $\beta (w,{w}_{*})$, defined in (32) with $\overline{\beta}=8$, $c=7$, $\nu =2$, ${\gamma}_{I}=1/12$, $\alpha =0.005$ and variable $\zeta $ as in (33) with $\psi =0.005$. We considered ${\zeta}_{0}=0.95$ (

**left**) and ${\zeta}_{0}=0.55$ (

**right**). The initial distribution is (29) with mass fractions (30). Bottom row: decline in vaccine efficacy due to the presence of a high number of infective people (

**left**) and the evolution of the Gini index (

**right**) for a variable infection rate $\beta (w,{w}_{*})$ as in (32) and vaccine effectiveness $\zeta \left(t\right)$ as in (33). We considered ${\lambda}_{S}=0.10$, ${\lambda}_{I}=0.07$, ${\lambda}_{V}=0.25$, ${\lambda}_{R}=0.15$ and $\overline{\beta}=8$, $c=7$, $\nu =2$ and $\psi =0.005$.

**Figure 8.**Test 3B. Top row: epidemic dynamics with wealth-dependent $\beta (w,{w}_{*})$, defined in (32) with $\overline{\beta}=8$, $c=7$, $\nu =2$, ${\gamma}_{I}=1/12$, ${\gamma}_{R}=1/180$, $\alpha =0.005$ and variable $\zeta $ as in (33) with $\psi =1.5\times {10}^{-4}$. We considered ${\zeta}_{0}=0.95$ (

**left**) and ${\zeta}_{0}=0.55$ (

**right**). The initial distribution is (29) with mass fractions (30). Bottom row: decline in vaccine efficacy due to the presence of a high number of infected people (

**left**) and evolution of the Gini index (

**right**). We considered ${\lambda}_{S}=0.10$, ${\lambda}_{I}=0.07$, ${\lambda}_{V}=0.25$, ${\lambda}_{R}=0.15$ and $\overline{\beta}=8$, $c=7$ and $\nu =2$.

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**MDPI and ACS Style**

Bernardi, E.; Pareschi, L.; Toscani, G.; Zanella, M. Effects of Vaccination Efficacy on Wealth Distribution in Kinetic Epidemic Models. *Entropy* **2022**, *24*, 216.
https://doi.org/10.3390/e24020216

**AMA Style**

Bernardi E, Pareschi L, Toscani G, Zanella M. Effects of Vaccination Efficacy on Wealth Distribution in Kinetic Epidemic Models. *Entropy*. 2022; 24(2):216.
https://doi.org/10.3390/e24020216

**Chicago/Turabian Style**

Bernardi, Emanuele, Lorenzo Pareschi, Giuseppe Toscani, and Mattia Zanella. 2022. "Effects of Vaccination Efficacy on Wealth Distribution in Kinetic Epidemic Models" *Entropy* 24, no. 2: 216.
https://doi.org/10.3390/e24020216